Laplace Transform Pairs for Bilateral
Transform pair
1
2
Signal
 t 
u t 
4
t n 1
u t 
n  1!
Transform
1
1
s
1
s
1
sn
3
 u  t 
5
t n 1

u  t 
n  1!
1
sn
Re( s )  0
1
sa
1
sa
1
s  a n
1
s  a n
Re(s )  a
e  sT
s
2
s  w02
All s
Re( s)  0
w0
2
s  w02
sa
s  a 2  w02
Re( s)  0
w0
Re( s )   a
6
e  at u t 
7
 e  at u  t 
8
t n 1  at
e u t 
n  1!
9
10
11
t n 1  at
e u  t 
n  1!
 t  T 
cos w0t u (t )
12
sin w0t u (t )
13
e cos w t u(t )
14
e sin w t u(t )
15
16

 at
0
 at
0
d n (t )
dt n
u  n t   u (t ) * * u (t )
u n t  
ROC
All s
Re( s )  0
Re( s )  0
Re( s )  0
Re( s)  a
Re(s )  a
Re( s )   a
Re( s )   a
s  a 2  w02
sn
All s
1
sn
Re( s)  0
Laplace Transform Property for Bilateral
Property
Linearity
Time Shifting
Shift in the s-Domain
Time Scaling
Conjugation
Convolution
Differentiation in the
time-Domain
Differentiation in the
s-Domain
Integration in the
Time Domain
Signal
x(t ) , x1 (t ) , x2 (t )
ax1 (t )  bx2 (t )
x(t  t0 )
Laplace Transform
X (s ) , X 1 ( s ) , X 2 ( s )
aX 1 ( S )  bX 2 ( S )
e  st0 X ( S )
X ( s  s0 )
1
s
X( )
a
a
e s0t x(t )
x(at )
x* (t )
x1 (t ) * x2 (t )
dx(t )
dt
 tx(t )
R
Scaled ROC
X * (S * )
X 1 (s) X 2 ( s)
SX ( S )
dX ( S )
dS
1
X (S )
S
t
 x( )d

Initial value theorem
x(0  )  lim SX ( S )
Final value theorem
lim x(t )  lim SX ( S )
ROC
R , R1 , R2
R1  R2
R
R
At least R1  R2
At least R
R
At least
R  Re( s )  0
s 
t 
s 0
Laplace Transform Property for Unilateral
Property
Linearity
Shift in the s-Domain
Time Scaling
Conjugation
Convolution
Differentiation in the
time-Domain
Differentiation in the
s-Domain
Integration in the
Time Domain
Signal
x(t ) , x1 (t ) , x2 (t )
ax1 (t )  bx2 (t )
e s0t x(t )
x(at )
x* (t )
x1 (t ) * x2 (t )
dx(t )
dt
d 2 x(t )
dt 2
 tx(t )
t
 x( )d

Initial value theorem
x(0  )  lim S ( S )
Final value theorem
lim x(t )  lim S ( S )
s 
t 
s 0
Laplace Transform
 (s ) , 1 ( s ) ,  2 ( s )
a 1 ( S )  b 2 ( S )
 ( s  s0 )
1
s
( )
a
a
 * (S )
1 ( s )  2 ( s )
S ( S )  x(0  )
S 2  ( S )  sx(0  )  x' (0  )
d ( S )
dS
1
 (S )
S
Table of z transforms 1
Table of z transforms 2
Property table:
Table: z transform of xk  m and xk  m